The rest of the paper is organized as follows. In Section , we
summarize the main results of the paper, including a new extremal entropy
inequality and its applications on the degraded vector Gaussian broadcast
channel with layered confidential messages. In Section , we
prove the generalized Costa EPI, following a perturbation approach via a
fundamental relationship between the derivative of mutual information and MMSE
estimate in linear vector Gaussian channels [1]}. In
Section , we derive the new extremal entropy inequality from
the generalized Costa EPI. The coding theorems for the degraded vector Gaussian
broadcast channel with layered confidential messages are proved in
Section  and Section . Finally, in
Section , we conclude the paper with some remarks.
